Dirichlet series
| L(s) = 1 | − 3-s + 2·4-s − 12·5-s + 3·7-s + 2·8-s + 6·9-s + 6·11-s − 2·12-s − 4·13-s + 12·15-s + 4·16-s − 2·17-s + 10·19-s − 24·20-s − 3·21-s − 3·23-s − 2·24-s + 36·25-s − 27-s + 6·28-s − 3·29-s − 10·31-s + 5·32-s − 6·33-s − 36·35-s + 12·36-s + 25·37-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 4-s − 5.36·5-s + 1.13·7-s + 0.707·8-s + 2·9-s + 1.80·11-s − 0.577·12-s − 1.10·13-s + 3.09·15-s + 16-s − 0.485·17-s + 2.29·19-s − 5.36·20-s − 0.654·21-s − 0.625·23-s − 0.408·24-s + 36/5·25-s − 0.192·27-s + 1.13·28-s − 0.557·29-s − 1.79·31-s + 0.883·32-s − 1.04·33-s − 6.08·35-s + 2·36-s + 4.10·37-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(11^{12} \cdot 13^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.91312\) |
| Root analytic conductor: | \(1.06857\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 11^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.7832956658\) |
| \(L(\frac12)\) | \(\approx\) | \(0.7832956658\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 11 | \( ( 1 - T + T^{2} )^{6} \) |
| 13 | \( 1 + 4 T + 17 T^{2} + 14 T^{3} - 12 T^{4} - 1104 T^{5} - 4629 T^{6} - 1104 p T^{7} - 12 p^{2} T^{8} + 14 p^{3} T^{9} + 17 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 2 | \( 1 - p T^{2} - p T^{3} + 3 T^{5} + 19 T^{6} + p T^{7} - 27 T^{8} - 17 p T^{9} - 21 T^{10} + 29 T^{11} + 183 T^{12} + 29 p T^{13} - 21 p^{2} T^{14} - 17 p^{4} T^{15} - 27 p^{4} T^{16} + p^{6} T^{17} + 19 p^{6} T^{18} + 3 p^{7} T^{19} - p^{10} T^{21} - p^{11} T^{22} + p^{12} T^{24} \) |
| 3 | \( 1 + T - 5 T^{2} - 10 T^{3} + 11 T^{4} + p^{3} T^{5} + 49 T^{6} + 23 p T^{7} - 190 T^{8} - 208 p T^{9} + 47 p T^{10} + 1115 T^{11} + 1147 T^{12} + 1115 p T^{13} + 47 p^{3} T^{14} - 208 p^{4} T^{15} - 190 p^{4} T^{16} + 23 p^{6} T^{17} + 49 p^{6} T^{18} + p^{10} T^{19} + 11 p^{8} T^{20} - 10 p^{9} T^{21} - 5 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 5 | \( ( 1 + 6 T + 36 T^{2} + 27 p T^{3} + 481 T^{4} + 1288 T^{5} + 3259 T^{6} + 1288 p T^{7} + 481 p^{2} T^{8} + 27 p^{4} T^{9} + 36 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 7 | \( 1 - 3 T - 17 T^{2} + 78 T^{3} + 73 T^{4} - 769 T^{5} - 11 T^{6} + 4647 T^{7} + 968 T^{8} - 4364 p T^{9} + 13961 T^{10} + 112933 T^{11} - 296357 T^{12} + 112933 p T^{13} + 13961 p^{2} T^{14} - 4364 p^{4} T^{15} + 968 p^{4} T^{16} + 4647 p^{5} T^{17} - 11 p^{6} T^{18} - 769 p^{7} T^{19} + 73 p^{8} T^{20} + 78 p^{9} T^{21} - 17 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 2 T - 64 T^{2} - 178 T^{3} + 2027 T^{4} + 6470 T^{5} - 44437 T^{6} - 131753 T^{7} + 849548 T^{8} + 1643836 T^{9} - 15968511 T^{10} - 9733593 T^{11} + 284268892 T^{12} - 9733593 p T^{13} - 15968511 p^{2} T^{14} + 1643836 p^{3} T^{15} + 849548 p^{4} T^{16} - 131753 p^{5} T^{17} - 44437 p^{6} T^{18} + 6470 p^{7} T^{19} + 2027 p^{8} T^{20} - 178 p^{9} T^{21} - 64 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 10 T - 5 T^{2} + 428 T^{3} - 1138 T^{4} - 7242 T^{5} + 31846 T^{6} + 78390 T^{7} - 447837 T^{8} - 1164664 T^{9} + 7411937 T^{10} + 9878996 T^{11} - 143378385 T^{12} + 9878996 p T^{13} + 7411937 p^{2} T^{14} - 1164664 p^{3} T^{15} - 447837 p^{4} T^{16} + 78390 p^{5} T^{17} + 31846 p^{6} T^{18} - 7242 p^{7} T^{19} - 1138 p^{8} T^{20} + 428 p^{9} T^{21} - 5 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 3 T - 73 T^{2} - 18 p T^{3} + 89 p T^{4} + 18037 T^{5} - 17187 T^{6} - 306933 T^{7} + 291234 T^{8} - 307744 T^{9} - 40315893 T^{10} + 44594697 T^{11} + 1483633367 T^{12} + 44594697 p T^{13} - 40315893 p^{2} T^{14} - 307744 p^{3} T^{15} + 291234 p^{4} T^{16} - 306933 p^{5} T^{17} - 17187 p^{6} T^{18} + 18037 p^{7} T^{19} + 89 p^{9} T^{20} - 18 p^{10} T^{21} - 73 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 + 3 T - 39 T^{2} - 332 T^{3} - 109 T^{4} + 9849 T^{5} + 42621 T^{6} - 33032 T^{7} - 1335138 T^{8} - 2907126 T^{9} + 21274646 T^{10} + 29638366 T^{11} - 263270952 T^{12} + 29638366 p T^{13} + 21274646 p^{2} T^{14} - 2907126 p^{3} T^{15} - 1335138 p^{4} T^{16} - 33032 p^{5} T^{17} + 42621 p^{6} T^{18} + 9849 p^{7} T^{19} - 109 p^{8} T^{20} - 332 p^{9} T^{21} - 39 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( ( 1 + 5 T + 5 p T^{2} + 718 T^{3} + 10921 T^{4} + 42369 T^{5} + 438894 T^{6} + 42369 p T^{7} + 10921 p^{2} T^{8} + 718 p^{3} T^{9} + 5 p^{5} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 37 | \( 1 - 25 T + 227 T^{2} - 862 T^{3} + 2511 T^{4} - 17777 T^{5} - 10367 T^{6} + 269768 T^{7} + 8006708 T^{8} - 43639662 T^{9} - 292434890 T^{10} + 2700439926 T^{11} - 11231672892 T^{12} + 2700439926 p T^{13} - 292434890 p^{2} T^{14} - 43639662 p^{3} T^{15} + 8006708 p^{4} T^{16} + 269768 p^{5} T^{17} - 10367 p^{6} T^{18} - 17777 p^{7} T^{19} + 2511 p^{8} T^{20} - 862 p^{9} T^{21} + 227 p^{10} T^{22} - 25 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 24 T + 148 T^{2} + 198 T^{3} + 4123 T^{4} - 111842 T^{5} + 235759 T^{6} + 1497091 T^{7} + 26077622 T^{8} - 164163914 T^{9} - 995029063 T^{10} + 128371115 T^{11} + 73816799112 T^{12} + 128371115 p T^{13} - 995029063 p^{2} T^{14} - 164163914 p^{3} T^{15} + 26077622 p^{4} T^{16} + 1497091 p^{5} T^{17} + 235759 p^{6} T^{18} - 111842 p^{7} T^{19} + 4123 p^{8} T^{20} + 198 p^{9} T^{21} + 148 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 8 T - 151 T^{2} + 1102 T^{3} + 14052 T^{4} - 79532 T^{5} - 1058966 T^{6} + 3935798 T^{7} + 68077509 T^{8} - 130956032 T^{9} - 3807272693 T^{10} + 1991672642 T^{11} + 180970516447 T^{12} + 1991672642 p T^{13} - 3807272693 p^{2} T^{14} - 130956032 p^{3} T^{15} + 68077509 p^{4} T^{16} + 3935798 p^{5} T^{17} - 1058966 p^{6} T^{18} - 79532 p^{7} T^{19} + 14052 p^{8} T^{20} + 1102 p^{9} T^{21} - 151 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( ( 1 + 10 T + 267 T^{2} + 2167 T^{3} + 30479 T^{4} + 4151 p T^{5} + 1893064 T^{6} + 4151 p^{2} T^{7} + 30479 p^{2} T^{8} + 2167 p^{3} T^{9} + 267 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 53 | \( ( 1 - 10 T + 162 T^{2} - 653 T^{3} + 4413 T^{4} + 37800 T^{5} - 144463 T^{6} + 37800 p T^{7} + 4413 p^{2} T^{8} - 653 p^{3} T^{9} + 162 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 + 4 T - 189 T^{2} - 602 T^{3} + 16338 T^{4} + 31734 T^{5} - 1130938 T^{6} - 1237904 T^{7} + 85433449 T^{8} + 65653840 T^{9} - 6560043281 T^{10} - 1659535836 T^{11} + 433965832587 T^{12} - 1659535836 p T^{13} - 6560043281 p^{2} T^{14} + 65653840 p^{3} T^{15} + 85433449 p^{4} T^{16} - 1237904 p^{5} T^{17} - 1130938 p^{6} T^{18} + 31734 p^{7} T^{19} + 16338 p^{8} T^{20} - 602 p^{9} T^{21} - 189 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 21 T - 69 T^{2} + 2586 T^{3} + 30551 T^{4} - 435587 T^{5} - 3687837 T^{6} + 30728974 T^{7} + 501799536 T^{8} - 34960040 p T^{9} - 40578292958 T^{10} + 33696701368 T^{11} + 3039523419348 T^{12} + 33696701368 p T^{13} - 40578292958 p^{2} T^{14} - 34960040 p^{4} T^{15} + 501799536 p^{4} T^{16} + 30728974 p^{5} T^{17} - 3687837 p^{6} T^{18} - 435587 p^{7} T^{19} + 30551 p^{8} T^{20} + 2586 p^{9} T^{21} - 69 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 21 T - 55 T^{2} + 2872 T^{3} + 16107 T^{4} - 366619 T^{5} - 2376877 T^{6} + 28271559 T^{7} + 343251034 T^{8} - 1946200298 T^{9} - 30101803053 T^{10} + 42695945413 T^{11} + 2380055001287 T^{12} + 42695945413 p T^{13} - 30101803053 p^{2} T^{14} - 1946200298 p^{3} T^{15} + 343251034 p^{4} T^{16} + 28271559 p^{5} T^{17} - 2376877 p^{6} T^{18} - 366619 p^{7} T^{19} + 16107 p^{8} T^{20} + 2872 p^{9} T^{21} - 55 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 3 T - 145 T^{2} + 154 T^{3} + 9357 T^{4} - 76659 T^{5} + 5181 T^{6} + 9273415 T^{7} - 40988312 T^{8} - 640901536 T^{9} + 3584950207 T^{10} + 17149435309 T^{11} - 267906892465 T^{12} + 17149435309 p T^{13} + 3584950207 p^{2} T^{14} - 640901536 p^{3} T^{15} - 40988312 p^{4} T^{16} + 9273415 p^{5} T^{17} + 5181 p^{6} T^{18} - 76659 p^{7} T^{19} + 9357 p^{8} T^{20} + 154 p^{9} T^{21} - 145 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( ( 1 + 13 T + 487 T^{2} + 4750 T^{3} + 93989 T^{4} + 693669 T^{5} + 9310278 T^{6} + 693669 p T^{7} + 93989 p^{2} T^{8} + 4750 p^{3} T^{9} + 487 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 79 | \( ( 1 + 4 T + 66 T^{2} + 546 T^{3} + 13463 T^{4} + 41230 T^{5} + 717404 T^{6} + 41230 p T^{7} + 13463 p^{2} T^{8} + 546 p^{3} T^{9} + 66 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( ( 1 + 8 T + 321 T^{2} + 2923 T^{3} + 51931 T^{4} + 440473 T^{5} + 5329268 T^{6} + 440473 p T^{7} + 51931 p^{2} T^{8} + 2923 p^{3} T^{9} + 321 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 89 | \( 1 + 9 T - 125 T^{2} + 1242 T^{3} + 20429 T^{4} - 269453 T^{5} + 444701 T^{6} + 21345379 T^{7} - 221304868 T^{8} + 109954236 T^{9} + 5001700447 T^{10} - 108237781475 T^{11} - 5833373813 T^{12} - 108237781475 p T^{13} + 5001700447 p^{2} T^{14} + 109954236 p^{3} T^{15} - 221304868 p^{4} T^{16} + 21345379 p^{5} T^{17} + 444701 p^{6} T^{18} - 269453 p^{7} T^{19} + 20429 p^{8} T^{20} + 1242 p^{9} T^{21} - 125 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 15 T - 90 T^{2} + 3643 T^{3} - 10364 T^{4} - 440399 T^{5} + 4794834 T^{6} + 10500051 T^{7} - 486535796 T^{8} + 1852502725 T^{9} + 23499881338 T^{10} - 171389457541 T^{11} + 48260420634 T^{12} - 171389457541 p T^{13} + 23499881338 p^{2} T^{14} + 1852502725 p^{3} T^{15} - 486535796 p^{4} T^{16} + 10500051 p^{5} T^{17} + 4794834 p^{6} T^{18} - 440399 p^{7} T^{19} - 10364 p^{8} T^{20} + 3643 p^{9} T^{21} - 90 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.53251942954580346336231903778, −4.51355719550414670296163478136, −4.43960237260513222389051821635, −4.20084959975490561141382221797, −4.12350825122778399163373494411, −4.08024199005731904581238150401, −3.97313159999043867047641607698, −3.92810172925826137548109869614, −3.83219945796032975779583328898, −3.71871175602936335143314039545, −3.70249348520767051997521891163, −3.63124628677789978180952243469, −3.26201357190505870710362315959, −3.08278293783152524923255937382, −2.81836438520194874214522193447, −2.73530454088216385348661350254, −2.59956127322273658982498386902, −2.26851154524745268271088286644, −2.23828278123206340757799759845, −1.98750232469379901628797334512, −1.72986734538677240644036629678, −1.45849150752071123007486358881, −1.34021097951153569216880452919, −0.72707416313253218939260345193, −0.71619804057158241587552221020, 0.71619804057158241587552221020, 0.72707416313253218939260345193, 1.34021097951153569216880452919, 1.45849150752071123007486358881, 1.72986734538677240644036629678, 1.98750232469379901628797334512, 2.23828278123206340757799759845, 2.26851154524745268271088286644, 2.59956127322273658982498386902, 2.73530454088216385348661350254, 2.81836438520194874214522193447, 3.08278293783152524923255937382, 3.26201357190505870710362315959, 3.63124628677789978180952243469, 3.70249348520767051997521891163, 3.71871175602936335143314039545, 3.83219945796032975779583328898, 3.92810172925826137548109869614, 3.97313159999043867047641607698, 4.08024199005731904581238150401, 4.12350825122778399163373494411, 4.20084959975490561141382221797, 4.43960237260513222389051821635, 4.51355719550414670296163478136, 4.53251942954580346336231903778
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.